Magnetic flux control in superconducting devices

ABSTRACT

A method includes generating a bias signal from a first device, and applying the bias signal to a second device, the first device having (a) a superconducting trace and (b) a superconducting quantum interference device (SQUID), in which a first terminal of the SQUID is electrically coupled to a first end of the superconducting trace, and a second terminal of the SQUID is electrically coupled to a second end of the superconducting trace, where generating the bias signal from the first device includes: applying a first signal Φ 1  to a first sub-loop of the SQUID; and applying a second signal Φ 2  to a second sub-loop of the SQUID, in which the first signal Φ 1  and the second signal Φ 2  are applied such that a value of a superconducting phase of the first device is incremented or decremented by a non-zero integer multiple n of 2π.

TECHNICAL FIELD

The present disclosure relates to magnetic flux control insuperconducting devices.

BACKGROUND

Quantum computing is a relatively new computing method that takesadvantage of quantum effects, such as superposition of basis states andentanglement to perform certain computations more efficiently than aclassical digital computer. In contrast to a digital computer, whichstores and manipulates information in the form of bits (e.g., a “1” or“0”), quantum computing systems can manipulate information using qubits.A qubit can refer to a quantum device that enables the superposition ofmultiple states (e.g., data in both the “0” and “1” state) and/or to thesuperposition of data, itself, in the multiple states. In accordancewith conventional terminology, the superposition of a “0” and “1” statein a quantum system may be represented, e.g., as α|0>+β|1>. The “0” and“1” states of a digital computer are analogous to the |0> and |1> basisstates, respectively of a qubit. The value |α|² represents theprobability that a qubit is in |0> state, whereas the value |β|²represents the probability that a qubit is in the |1> basis state.

SUMMARY

In general, some aspects, the subject matter of the present disclosurecan be embodied in methods that include generating a bias signal from afirst device, and applying the bias signal to a second device, the firstdevice including (a) a superconducting trace and (b) a superconductingquantum interference device (SQUID) having at least three non-linearinductor junctions coupled in parallel, in which a first terminal of theSQUID is electrically coupled to a first end of the superconductingtrace, and a second terminal of the SQUID is electrically coupled to asecond end of the superconducting trace to form a loop, in whichgenerating the bias signal from the first device includes: applying afirst time-varying magnetic flux Φ₁ to a first sub-loop of the SQUID;and applying a second time-varying magnetic flux Φ₂ to a second sub-loopof the SQUID, in which the first time-varying magnetic flux Φ₁ and thesecond time-varying magnetic flux Φ₂ are applied such that a value of asuperconducting phase of the first device is incremented or decrementedby a non-zero integer multiple n of 2π.

Implementations of the methods can include one or more of the followingfeatures. For example, in some implementations, the output stateincludes an effective phase offset of a current through the firstdevice.

In some implementations, the output state includes an effective fluxthrough the first device.

In some implementations each of a maximum magnitude of the firsttime-varying magnetic flux Φ₁ and a maximum magnitude of the secondtime-varying magnetic flux Φ₂ is less than the flux quantum Φ₀.

In some implementations, a ratio, Φ₁/Φ₀, of the first time-varyingmagnetic flux Φ₁ to the flux quantum Φ₀, and a ratio, Φ₂/Φ₀, of thesecond time-varying magnetic flux Φ₂ to the flux quantum Φ₀ trace a patharound a point where a current value through the first device is 0. Theratio Φ₁/Φ₀ and the ratio Φ₂/Φ₀ can be, e.g., approximately equal to 1/3at the point where the current through the first device is 0. Theinteger multiple n can be incremented when the ratio Φ₁/Φ₀ and the ratioΦ₂/Φ₀ trace the path along a first direction around the point where thecurrent through the first device is 0, or decremented when the ratioΦ₁/Φ₀ and the ratio Φ₂/Φ₀ trace the path along a second direction thatis opposite to the first direction. The path can be a closed-loop path.

In some implementations, a ratio, Φ₁/Φ₀, of the first time-varyingmagnetic flux Φ₁ to the flux quantum Φ₀, and a ratio, Φ₂/Φ₀, of thesecond time-varying magnetic flux Φ₂ to the flux quantum Φ₀ trace a paththrough a point where an effective phase offset of the current throughthe first device is 0. The path can be a closed-loop path.

In some implementations, applying the first time-varying magnetic fluxand the second time-varying magnetic flux includes changing the phaseassociated with each Josephson junction of the SQUID by 2π.

In some implementations, the method includes cooling the first device tobelow the superconducting critical temperature of a superconductingmaterial in the superconducting trace.

In some implementations, the first time-varying magnetic flux Φ₁ and thesecond time-varying magnetic flux Φ₂ are sequentially applied. The firsttime-varying magnetic flux Φ₁ and the second time-varying magnetic fluxΦ₂ can overlap in time.

In general, in some other aspects, the subject matter of the presentdisclosure can be embodied in devices that include: a first devicehaving a superconducting trace, and a superconducting quantuminterference device (SQUID) having at least three non-linear inductorjunctions coupled in parallel, in which a first terminal of the SQUID iselectrically coupled to a first end of the superconducting trace, and asecond terminal of the SQUID is electrically coupled to a second end ofthe superconducting trace to form a loop; and a second device arrangedproximate to the first device, in which a state of the second device iscontrollable by a bias generated by the first device.

Implementations of the devices can include one or more of the followingfeatures. For example, in some implementations, the second device is aqubit.

In some implementations, the second device is a qubit coupler element.

In general, in some other aspects, the subject matter of the presentdisclosure can be embodied in systems that include: multiple cellsarranged in an array of M rows by N columns, M being an integer numbergreater than or equal to 1, N being an integer number greater than orequal to 2, in which each cell of the multiple cells includes acorresponding magnetic flux control device having: a superconductingtrace, and a superconducting quantum interference device (SQUID) havingat least three non-linear inductor junctions coupled in parallel, inwhich a first terminal of the SQUID is electrically coupled to a firstend of the superconducting trace, and a second terminal of the SQUID iselectrically coupled to a second end of the superconducting trace toform a loop.

Implementations of the systems can include one or more of the followingfeatures. For example, in some implementations, each cell of themultiple cells further includes a corresponding second device positionedproximate to the magnetic flux control device. In some implementations,for each cell of the multiple cells, the second device includes a qubit.

In some implementations, for each cell of the multiple cells, the seconddevice includes a qubit coupler element.

In some implementations, the systems further include: M first controllines, in which each first control line of the M first control linesextends along a corresponding row of the array and is couplable to eachmagnetic flux control device within the corresponding row; and N secondcontrol lines, in which each second control line of the N control linesextends along a corresponding column of the array is couplable to eachmagnetic flux control device within the corresponding column. Thesystems can further include: a row select generator coupled to the Mfirst control lines, the row select generator being configured toprovide a unique corresponding signal to each first control line of theM first control lines; and a column select generator coupled to the Nsecond control lines, the column select generator being configured toprovide a unique corresponding signal to each second control line of theN second control lines.

In general, in some other aspects, the subject matter of the presentdisclosure can be embodied in methods of operating a multi-level memorydevice that includes (a) a superconducting trace and (b) asuperconducting quantum interference device (SQUID) having at leastthree non-linear inductor junctions coupled in parallel, in which afirst terminal of the SQUID is electrically coupled to a first end ofthe superconducting trace, and a second terminal of the SQUID iselectrically coupled to a second end of the superconducting trace toform a loop, the methods including: applying a first time-varyingmagnetic flux Φ₁ to a first sub-loop of the SQUID, and applying a secondtime-varying magnetic flux Φ₂ to a second sub-loop of the SQUID, toplace the multi-level memory device in a first memory state.

Implementations of the methods can include one or more of the followingfeatures. For example, in some implementations, applying the firsttime-varying magnetic flux Φ₁ and applying a second time-varyingmagnetic flux Φ₂ causes an output state of the multi-level memory deviceto change by a non-zero integer n.

In some implementations, the methods include applying a thirdtime-varying magnetic flux Φ₃ to the first sub-loop of the SQUID, andapplying a fourth time-varying magnetic flux Φ₄ to the second sub-loopof the SQUID, to place the multi-level memory device in a second memorystate, wherein the second memory state is different than the firstmemory state.

Particular implementations of the subject matter described here canrealize one or more of the following advantages. For example, in someimplementations, the magnetic flux control devices of the presentdisclosure are capable of dissipating substantially little power, andthus providing an advantageous option as control devices for quantumcomputing circuit elements. For example, in certain implementations, themagnetic flux control devices of the present disclosure can dissipate afactor of 10³-10⁵ less power than CMOS-based or SFQ-based controldevices. Because the power dissipation, and thus heat generation, of themagnetic flux control devices is so low, the control devices can, incertain implementations, even be arranged on the same chip as thequantum computing circuit elements without substantially increasing thelocal chip temperature and/or without causing transitions to undesiredenergy states. In some implementations, the magnetic flux controldevices of the present disclosure can be operated as logical AND gates,where a HIGH output (e.g., a change in the magnetic flux control deviceoutput state) results only if both inputs meet a predetermined criteria,and a LOW output (e.g., no change in the magnetic flux control deviceoutput state) results if any of the inputs do not meet the predeterminedcriteria. Multiple magnetic flux control devices can be arranged in anarray or matrix configuration to provide multiplexed control formultiple quantum computing circuit elements. In some implementations,the magnetic flux control devices of the present disclosure can beoperated as multi-level memory devices.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features andadvantages will be apparent from the description, the drawings, and theclaims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic illustrating an example of a magnetic flux controldevice and a qubit.

FIG. 2 is a schematic illustrating an example of a superconductingquantum interference device (SQUID) that can be used in a magnetic fluxcontrol device.

FIG. 3A is a schematic illustrating an example of first time-varyingmagnetic flux signal applied to a first sub-loop of a magnetic fluxcontrol device and a second time-varying magnetic flux signal applied tosecond sub-loop of the magnetic flux control device.

FIG. 3B is a plot illustrating the magnitude of the first magnetic fluxwaveform versus the magnitude of the second magnetic flux waveform ofFIG. 3 for different points in time.

FIG. 4 is a plot illustrating the magnitude of a first magnetic fluxsignal normalized by flux quantum Φ₀ versus the magnitude of a secondmagnetic flux signal normalized by flux quantum Φ₀ overlaid against aheat map depicting the ratio of the magnitude of effective criticalcurrent I_(0e) to critical current I₀ through a magnetic flux controldevice.

FIG. 5 is a heat map plot illustrating a phase of current through amagnetic flux control device as a function of magnetic flux (Φ_(L)/Φ₀)applied to a first loop of the magnetic flux control device and ofmagnetic flux (Φ_(R)/Φ₀) applied to a second loop of the magnetic fluxcontrol device.

FIG. 6 is a schematic illustrating an example of a magnetic flux controldevice.

FIG. 7 is a schematic illustrating an example of a controllable matrixarray of magnetic flux control elements.

FIG. 8 is a table that illustrates an example of different output statesthat can be established by a multi-level memory

DETAILED DESCRIPTION

Quantum computing entails coherently processing quantum informationstored in the quantum bits (qubits) of a quantum computer.Superconducting quantum computing is a promising implementation ofquantum computing technology in which quantum computing circuit elementsare formed, in part, from superconducting materials. Superconductingquantum computers are typically multilevel systems, in which the twolowest energy levels are used as the computational basis. It ispreferably to operate quantum circuit elements (e.g., quantum computingcircuit elements) with low energy loss and dissipation (e.g., thequantum computing circuit elements exhibit a high quality factor, Q).Low energy loss and dissipation may help to avoid, e.g., quantumdecoherence and/or transitions to other undesired energy states.

One source of loss and dissipation is heat generated from controlelements, such as qubit control elements. Qubit control elements canhave different functions. For example, in some cases, a qubit controlelement provides control signals (e.g., flux biases) to tilt/perturb thedouble well potential during operation of the qubit. In some cases, aqubit control element provides control signals to adjust the magnitudeof the barrier between the potential wells during operation of the qubitor to change the operating frequency of the qubit. Additional controlfunctions are also possible. Another example of a control elementincludes the qubit coupler control element, which provides, e.g.,control signals to modify the coupling strength between qubits and qubitcoupler elements. Other control elements also are possible.

In general, the control elements can be sources of relatively high powerdissipation, and thus heat generation. When such a control element isarranged on the same chip as the superconducting quantum computingcircuit elements (e.g., qubits), the level of local energy dissipationmay lead to heating that can be significant relative to the lowtemperatures (e.g., ˜20 mK) at which the quantum computing circuitelements need to be maintained to achieve superconductivity. The highlevel of energy dissipation and subsequent heat generation therefore canrender it difficult to achieve the low temperatures at which the devicesare required to operate. Even for low power alternatives, such as singleflux quantum (SFQ) digital logic devices, the level of power dissipationstill may be too high.

The present disclosure is directed to magnetic flux control devicescapable of dissipating substantially little power, and thus providing anadvantageous option as control devices for quantum computing circuitelements. For example, in certain implementations, the magnetic fluxcontrol devices of the present disclosure can dissipate a factor of10³-10⁵ less power than CMOS-based or SFQ-based control devices.Accordingly, because the power dissipation, and thus heat generation, ofthe magnetic flux control devices is so low, the control devices can, incertain implementations, even be arranged on the same chip as thequantum computing circuit elements without substantially increasing thelocal chip temperature and/or without causing transitions to undesiredenergy states.

Changing an output state of the magnetic flux control device includessequentially applying multiple separate time-varying input signals(e.g., magnetic flux signals Φ₁, Φ₂) to the magnetic flux controldevice, in which the time-varying input signals meet predeterminedcriteria. The predetermined criteria can include causing asuperconducting phase of the magnetic flux control device to change by2π, where n is a non-zero integer number. More specifically, thepredetermined criteria can include applying the input signals (e.g.,magnetic flux signals e.g., Φ₁ and Φ₂) such that a ratio of the inputsignals to a flux quantum Φ₀ (e.g., Φ₁/Φ₀ and Φ₂/Φ₀) trace a path, e.g.,a closed-loop path, around a critical point. The critical point cancorrespond to where a current I through the device is 0. The criticalpoint can occur, e.g., when Φ₁/Φ₀=Φ₂/Φ₀≈1/3. In some implementations,the predetermined criteria can include applying the input signals (e.g.,magnetic flux signals Φ₁ and Φ₂) such that a ratio of the input signalsto a flux quantum Φ₀ (e.g., Φ₁/Φ₀ and Φ₂/Φ₀) trace a path, e.g., aclosed-loop path, through a point where an effective phase offset of acurrent I of the magnetic flux control device is 0.

When the applied flux meets the predetermined criteria, the output stateof the magnetic flux control device changes. For example, an effectiveflux (as distinguished from the fields that are applied to the magneticflux control device) through the magnetic flux control device changes bya non-zero integer number n of flux quantum Φ₀. In contrast, when theapplied flux signals do not meet the predetermined criteria, the outputstate of the magnetic flux control device does not change. For example,the effective flux through the magnetic flux control device does notchange. Accordingly, in some implementations, the magnetic flux controldevice of the present disclosure can be operated as a logical AND gate,where a HIGH output (e.g., a change in the magnetic flux control deviceoutput state) results only if both inputs meet a predetermined criteria,and a LOW output (e.g., no change in the magnetic flux control deviceoutput state) results if any of the inputs do not meet the predeterminedcriteria. Multiple magnetic flux control devices can be arranged in anarray or matrix configuration to provide multiplexed control formultiple quantum computing circuit elements.

In some implementations, the magnetic flux control device of the presentdisclosure can be operated as a multi-level memory device. That is,since the output state (e.g., the effective flux or the superconductingphase) of the magnetic flux control device is proportional to n, eachinteger change in n can correspond to a different memory state. As willbe understood from present disclosure, other applications and advantagesof the magnetic flux control device are also possible.

FIG. 1 is a schematic illustrating an example of a magnetic flux controldevice 100 and a second device 150, such as a qubit. Both the controldevice 100 and the second device 150 can be formed on a dielectricsubstrate, such as silicon or sapphire. In the present example, thesecond device 150 is a two-junction qubit including a superconductingtrace 114 arranged in a loop or ring-like structure. The superconductingtrace 114 is coupled in two separate locations to correspondingnon-linear inductor junctions 110, 112. In the examples disclosedherein, a non-linear inductor junction can include a Josephson junction,which is a quantum-mechanical device made of two superconductingelectrodes separated by an insulating barrier. An example of a Josephsonjunction includes a layer of aluminum oxide sandwiched between a firstlayer of aluminum and a second layer of aluminum. Superconducting trace114 can be formed from material capable of achieving superconductivity.For example, trace 114 can be formed from aluminum or niobium. Thesecond device 150 is not limited to the configuration shown in FIG. 1and can include other designs, including other qubit designs, instead.For example, in some implementations, second device 150 includes a phasequbit, a charge qubit or a flux qubit. Examples of flux qubits that canbe used include, e.g., a fluxmon qubit, a gmon qubit, or an x-mon qubit.The qubit can include two, three, four or other numbers of Josephsonjunctions. Other qubit configurations are also possible. Second device150 can include devices other than qubits. For example, in someimplementations, second device 150 includes a qubit coupler element. Aqubit coupler element can include, for example, a resonator element,such as a superconducting resonator element.

Second device 150 is arranged proximately to the magnetic flux controldevice 100 so that second device 150 is exposed to a bias signalgenerated by the magnetic flux control device 100. The bias signalgenerated by the magnetic flux control device 100 can include a magneticflux M generated by magnetic flux control device 100.

The magnetic flux control device 100 includes a superconducting trace102, in which one end of the superconducting trace 102 is connected to afirst terminal 106 of a superconducting quantum interference device(SQUID) 104 and a second end of the superconducting trace 102 isconnected to a second terminal 108 of the SQUID 104 to form a ring-likestructure or loop 101. Superconducting trace 102 can be formed frommaterial capable of achieving superconductivity. For example, trace 102can be formed from aluminum or niobium. Operation of the magnetic fluxcontrol device 100 includes cooling the magnetic flux control device 100below the superconducting critical temperature of the superconductingmaterials in the device 100, and then applying multiple input signals(e.g., time-varying magnetic flux signals) to the SQUID 104, such that acurrent I is generated in the superconducting trace 102. The current Itraveling through the superconducting trace 102 gives rise to themagnetic flux M that can inductively couple to the second device 150positioned proximately to the magnetic flux control device 105.

The current I can be expressed as I≈nΦ₀/L, where n is an integer number,L is the dominant inductance of the loop (e.g., the inductance of theloop formed by the superconducting trace 102 and the SQUID 104), and Φ₀is the flux quantum, which can be expressed as h/2e, where h is Planck'sconstant and e is the charge on an electron.

A state of the second device 150 is controllable by the bias signalgenerated by the magnetic flux control device 100. That is, based on theparticular magnetic flux signals applied to the SQUID 104, it ispossible, in certain implementations, to change the value of n and, inturn, alter the current I or an effective magnetic flux M (where M canbe expressed as approximately M≈nΦ₀) produced by the device 100 andreceived by second device 150. Accordingly, in some implementations, themagnetic flux control device 100 serves as a control element for anexternal device, such as second device 150. For example, the magneticflux control device 100 can be used to modify the tilt of the qubit'spotential well, modify the magnitude of the qubit's potential wellbarrier magnitude based on the value of n, or change the qubit'sfrequency.

Although second device 150 is shown in the example as a superconductingqubit, other devices can be used instead. For example, in someimplementations, the bias signal generated by the magnetic flux controldevice 100 can be used to control a qubit coupler element. That is, thebias (e.g., the magnetic flux or the current signal) generated by device100 can be varied to cause a corresponding change in the couplingstrength of the qubit coupler element. Alternatively, the bias generatedby device 100 can be varied to cause a corresponding change in frequencyof the qubit coupler element.

FIG. 2 is a schematic illustrating an example of a SQUID 200 that can beused as element 104 in the magnetic flux control device 100. SQUID 200is a three-junction SQUID, meaning that SQUID 200 includes threenon-linear inductor junctions 210, 220, and 230. The non-linear inductorjunctions 210, 220 and 230 are directly electrically coupled in parallelto one another at a first terminal 206 and a second terminal 208. Forexample, the non-linear inductor junctions 210, 220, and 230 can beelectrically coupled in parallel through a superconducting trace 202.Superconducting trace 202 can be formed from material capable ofachieving superconductivity. For example, trace 202 can be formed fromaluminum or niobium. Terminal 206 is equivalent to terminal 106 andterminal 208 is equivalent to terminal 108, as shown in FIG. 1.

By coupling together the junctions 210, 220 and 230 in this manner, twoloops (sub-loop 240 and sub-loop 250) are provided. Sub-loop 240 isdefined by the current path through non-linear inductor junction 210,superconducting trace 202 and non-linear inductor junction 220. Sub-loop250 is defined by the current path through non-linear inductor junction220, superconducting trace 202 and non-linear inductor junction 230.When SQUID 200 is coupled to a superconductor trace, such as trace 102in the example of FIG. 1, the sub-loops 240, 250 correspond to sub-loopsof the larger ring defined by the current path through trace 102 and theSQUID 200.

Operation of magnetic flux control device 100 will be described hereinusing SQUID 200 as an example of element 104, though other SQUID designsare also possible for use as element 104. For example, though SQUID 200is illustrated in FIG. 2 as having three-non-linear inductor junctionsin parallel, SQUIDS with more than three non-linear inductor junctionscan be used instead. In some implementations, SQUIDS having four, five,six or other numbers of non-linear inductor junctions coupled inparallel can be used instead. In cases where the SQUID includes morethan three non-linear inductor junctions/more than two sub-loops, theoutput state of the magnetic flux control device still can be changed byapplying a corresponding flux signal to each sub-loop, in which the fluxsignals meet predetermined criteria, as explained herein.

During operation of the magnetic flux control device 100, two inputsignals, e.g., time-varying magnetic flux signals Φ_(L) and Φ_(R), areapplied to sub-loop 240 and sub-loop 250 of SQUID 200, respectively. Thefield can be provided by positioning separate inductors (e.g.,superconducting traces) adjacent to each of sub-loops 240, 250 andproviding current through the inductors to generate a magnetic field.The application of the time-varying fluxes to the two loops results ingeneration of a current through each non-linear inductor junction ofSQUID 200. The current through junction 210 can be expressed as I₀sin(δ−2πΦ_(L) Φ₀), where I₀ is the critical current through the junction(the value of which can be set by the junction fabrication process), δis a phase difference across the device, and Φ₀ is the flux quantum. Thecurrent through junction 230 can be expressed as I₀ sin(δ+2πΦ_(L) Φ₀).The current through the center junction 220 can be expressed as I₀sin(δ).

The current I (δ) through SQUID 200 (i.e., the current passing fromterminal 208 to terminal 206) then can be expressed as a combination ofthe currents through each non-linear inductor junction or:

$\quad\begin{matrix}{{I(\delta)} = {{I_{0}\mspace{11mu}{\sin(\delta)}} + {I_{0}\mspace{11mu}{\sin\left( {\delta + {2{\pi\Phi}_{L}\Phi_{0}}} \right)}} + {I_{0}\mspace{11mu}{\sin\left( {\delta - {2{\pi\Phi}_{L}\Phi_{0}}} \right)}}}} \\{= {{Im}\left\{ {I_{0}{e^{i\;\delta}\left( {1 + e^{i\; 2{\pi\Phi}\; R\text{/}{\Phi 0}} + e^{{- i}\; 2{\pi\Phi}\; L\text{/}{\Phi 0}}} \right)}} \right.}} \\{{= {{I_{0e}\left( {\Phi_{R},\Phi_{L}} \right)}\mspace{11mu}{\sin\;\left\lbrack {\delta + {\phi\mspace{11mu}\left( {\Phi_{R},\Phi_{L}} \right)}} \right\rbrack}}},}\end{matrix}$

Where I_(0e) is the effective critical current of the SQUID 200 as afunction of Φ_(R) and Φ_(L), and ϕ is the effective phase offset of thecurrent I (δ) as a function of Φ_(R) and Φ_(L).

The foregoing equations apply to the limit where the inductance of thesub-loops is small (e.g., where the Josephson inductance of the magneticflux control device is less than the loop inductance L of the magneticflux control device). FIG. 3A is a schematic illustrating an examplewaveform of a first time-varying magnetic flux signal 300 (depicted asΦ_(L) in FIG. 3A) applied to the first sub-loop 240 of the SQUID 200 ofFIG. 2 and a second time-varying magnetic flux signal 302 (depicted asΦ_(R) in FIG. 3A) applied to the second sub-loop 250 of the SQUID 200 ofFIG. 2. It is possible to change the output state of the magnetic fluxcontrol device 100 by varying the first time-varying magnetic fluxsignal 300 and the second time-varying magnetic flux signal 302according to predetermined criteria. In particular, it is possible tocause the effective phase offset ϕ, and thus the superconducting phase,of a current I through the SQUID 200 to change by 2πn, where n is aninteger number. As explained herein, the current I through the magneticflux device also can be expressed as I=nΦ₀/L. Since each change ineffective phase offset ϕ by 2π is equivalent to changing n by 1, thecurrent I, which is proportional to n, also changes. That is, changingthe effective phase offset by 2π changes the effective flux through theloop of the magnetic flux control device (where, e.g., the loop isdefined by the trace 102 and the SQUID element 104) by Φ₀, which changesthe loop current by approximately Φ₀/L. Thus, an integer change in n canbe achieved by applying input signals to the SQUID 200 to change thephase associated with the Josephson junctions by 2π. In contrast, whenthe signals (e.g., the magnetic flux signals) applied to the firstsub-loop 240 and to the second sub-loop 250 do not meet thepredetermined criteria, the output state of the magnetic flux controldevice does not change.

The waveforms 300 and 302 are examples of magnetic flux signals thatmeet predetermined criteria for changing the phase of the current I. Thewaveforms 300, 302 are divided into four separate periods of time, T=1,2, 3 and 4. During the first time period, T=1, the magnitude of bothflux 300 and flux 302 have the same initial value, e.g., both fluxes arezero. During the second time period, T=2, the magnitude of flux 300linearly increases to a maximum value, such as the flux quantum Φ₀,while the magnitude of flux 302 does not change from its initial value.For instance, the magnitude of flux 302 remains zero. During the thirdtime period, T=3, the magnitude of flux 300 begins to linearly decreasetoward a final value, whereas the magnitude of flux 302 begins tolinearly increase to a maximum value, such as Φ₀. In someimplementations, the final value of flux 300 can be equal to the initialvalue, e.g., zero, though other final values are also possible. Duringthe fourth time period, T=4, the magnitude of flux 300 does not change,whereas the magnitude of flux 302 begins to linearly decrease to a finalvalue. For instance, in some implementations, the final value of flux302 can be equal to the initial value, e.g., zero, though other finalvalues are also possible.

FIG. 3B is a plot of the magnitude of the first magnetic flux signal 300(depicted as Φ_(L)) against the magnitude of the second magnetic fluxsignal 302 (depicted as i) for the different time periods T=1, 2, 3, and4 shown in FIG. 3A. As illustrated in FIG. 3B, the magnitudes of themagnetic flux signals plotted against one another trace a closed-looppath, have maximum values of Φ₀. By applying the magnetic flux signalsin the foregoing manner, the phase across the magnetic flux controldevice changes by 2π, that is, the effective flux through the magneticflux control device changes by Φ₀.

The criteria required for the applied signals to cause a change in theeffective phase offset ϕ by 2π can be understood by referring to FIGS. 4and 5. FIG. 4 is a plot illustrating the magnitude of the first magneticflux waveform 300 (represented in FIG. 4 as Φ_(L)) normalized by Φ₀versus the magnitude of the second magnetic flux waveform 302(represented in FIG. 4 as Φ_(R)) normalized by Φ₀ and overlaid against aheat map depicting the ratio of the magnitude of effective criticalcurrent I_(0e) to critical current I₀ through the magnetic flux controldevice 100. The values of Φ_(L)/Φ₀ and Φ_(R)/Φ₀ trace a closed-loop path400 around the heat map. For each position along the closed-loop path400, the ratio I_(0e)/I₀ is determined by the heat map at that position.To obtain a change in the effective phase offset ϕ by 2π, the appliedfluxes trace a closed-loop path around a critical point 402 on the heatmap. In the present example, the critical point 402 corresponds to theposition at which the ratio I_(0e)/I₀ equals 0. Thus, when the magneticfluxes applied to the first and second sub-loops of the SQUID 200complete a closed-loop path around the critical point 402, the effectivephase offset ϕ across the magnetic flux control device 100 changes by2π, leading to a change in n by 1. In the present example, the criticalpoint 402 occurs at values Φ_(L)/Φ₀ and Φ_(R)/Φ₀ that are approximatelyequal to 1/3. The value at which the critical point occurs can changedepending on the inductance of the magnetic flux control device loop(determined, e.g., by the trace 102 and SQUID element 104 shown in FIG.1). When the magnetic fluxes are applied such that the closed-loop path400 is traced in a counter-clockwise manner, the phase increases by 2πfor each full revolution. For example, if the overlapping magnetic fluxsignals shown in FIG. 3A is repeated a second time, the effective phaseoffset across the magnetic flux device will increase by another 2π and nincreases by 2.

The predetermined criteria for achieving a 2π phase change across themagnetic flux control device 100, and thus an integer change in theeffective flux by Φ₀, alternatively can be understood from FIG. 5. FIG.5 is a plot illustrating the magnitude of the first magnetic fluxwaveform 300 (represented in FIG. 5 as Φ_(L)) normalized by Φ₀ versusthe magnitude of the second magnetic flux waveform 302 (represented inFIG. 5 as Φ_(R)) normalized by Φ₀ and overlaid against a heat mapdepicting the phase offset ϕ (Φ_(R), Φ_(L)) across the magnetic fluxcontrol device 100 (where the phase offset is represented in degrees).The values of Φ_(L)/Φ₀ and Φ_(R)/Φ₀ trace a closed-loop path 500 aroundthe heat map. For each position along the closed-loop path 500, theeffective phase offset across the magnetic flux control device isdetermined by the heat map at that position. To obtain a change in thephase offset ϕ (Φ_(R), Φ_(L)) by 2π, the closed-loop path passes througha location (e.g., the line 502) on the heat map where the phase goesfrom negative values through zero to positive values (or positive valuesthrough zero to negative values, depending on the direction the path istraced). Again, when the magnetic fluxes are applied such that theclosed-loop path 500 is traced in a counter-clockwise manner, theeffective phase offset increases by 2π for each full revolution.

As shown in FIG. 3A, the waveforms can be applied sequentially to eachsub-loop of the SQUID 200 and can overlap in time. The specifictrajectory traced by Φ_(L)/Φ₀ and Φ_(R)/Φ₀ is not critical to achievingthe 2 π phase change, so long as the predetermined criteria are met whenapplying the magnetic flux signals to the sub-loops of the SQUID 200.For example, the magnetic flux waveforms 300 and 302 are shown in FIG. 3as phase-shifted saw-tooth patterns. Other waveforms can be usedinstead. For instance, the waveforms can have a sinusoidal or othershape. In some implementations, the change from the initial value to themaximum value in the first waveform and/or the second waveform can beobtained using a step-increase function. Alternatively, in someimplementations, the change from the maximum value to the final value inthe first waveform and/or the second waveform can be obtained using astep-decrease function. In some implementations, the maximum valueachieved by the first waveform is different from the maximum valueachieved by the second waveform. To achieve a change in n by 1, however,the maximum magnitude of the flux (Φ_(L) and Φ_(R)) applied to eachsub-loop of the SQUID 200 should not exceed the flux quantum value Φ₀.In some implementations, the magnitude of the flux applied to eachsub-loop of the SQUID 200 can exceed the flux quantum Φ₀ such thatadditional critical points are encircled, leading to a change in n by aninteger number greater than 1.

In some implementations, the waveform applied to the first sub-loop 240of the SQUID 200 is not identical to the waveform applied to the secondsub-loop 250 of the SQUID 200. In some implementations, initial and/orfinal values of the first waveform and/or of the second waveform arenon-zero. The waveforms shown in FIG. 3A depict the magnitude of thefluxes increasing from a relatively low initial value to a highermaximum value and back to a low final value. However, in someimplementations, the change in flux magnitude can be inverted. Forexample, the flux magnitude of each waveform can start at a relativelyhigh initial value, decrease to a lower minimum value, and then increaseto a higher final value. Such waveforms, when applied to the SQUID 200,can result in the triangle-shaped path shown in FIG. 3B being flippedlaterally about a vertical axis.

The phase offset between the maximum value of the first waveform and themaximum value of the second waveform also can vary, so long as thewaveforms overlap to achieve the predetermined criteria that allows theeffective phase offset ϕ to change by 2π. For example, although FIG. 3Ashows the first waveform 300 reaching a maximum value before the secondwaveform 302 reaches a maximum value, the application of the waveformscan be reversed so that the second waveform 302 reaches a maximum valuebefore the first waveform 300 reaches a maximum value. In this case, thetrajectory traced by Φ_(L)/Φ₀ and Φ_(R)/Φ₀ would proceed in the oppositedirection than that shown in FIG. 3B. That is, rather than proceedingaround the closed-loop path in a counter-clockwise manner as shown inFIG. 3B, the trajectory traced by Φ_(L)/Φ₀ and Φ_(R)/Φ₀ would proceed ina clockwise manner. Reversing the trajectory traced by Φ_(L)/Φ₀ andΦ_(R)/Φ₀ causes the effective phase offset ϕ to decrease by integermultiples of 2π each time. With each 2π decrease in phase, the value ndecreases by 1. Thus, the sign of the change in n is determined by thedirection in which the trajectory of the magnetic fluxes applied to theSQUID sub-loops encircle the critical point. The absolute value for n isdetermined by the number of consecutive times that the input signalsapplied to the SQUID of the magnetic flux control device cause anincrease or decrease in the effective phase offset by 2π. For example,if the effective phase offset is increased by 6π, then n=3. In anotherexample, if the effective phase offset is decreased by 6π, then n=−3. Inanother example, if the effective phase offset is increased by 6π, andsubsequently decreased by 2π, then n=2.

To avoid coupling to the resonance frequencies of the sub-loops of SQUID200, the flux applied to each sub-loop should be associated with acorresponding frequency that is less than the resonant frequency of thesub-loop (e.g., in some cases the resonant frequency of the SQUIDsub-loop can be in the range of 10˜50 GHz). In some implementations,reducing the frequency of the applied flux signal can reduce powerdissipation of the device. This is because as the frequency of operationis reduced, the voltage produced by the device decreases. With lowervoltage, the power dissipated through operation of the magnetic fluxcontrol device decreases.

As explained with respect to FIG. 1, the current I through the magneticflux device 100 can be expressed as I=nΦ₀/L, where L is the inductanceof the superconducting trace 102. To achieve even finer changes in I,the inductance of the superconducting trace 102 can be increased byadding additional inductive material to the device 100. For example,FIG. 6 is a schematic illustrating an example of a magnetic flux controldevice 600 in which the effective inductance L of the device 600 ismodified by adding an additional loop of superconducting trace 616 andan additional 3-junction element 620 (e.g., a 3-junction SQUID, such asthe SQUID 200 shown in FIG. 2) coupled to trace 602. The first portionof the device 600 containing the trace 602 and SQUID 604 is identical tothe device 100 shown in FIG. 1 and is associated with an inductance L₁.The second portion of the device 600 includes the second trace 616 andadditional 3-junction element 620 is associated with inductance L₂ andis coupled with trace 602. Through the addition of trace 616 and the3-junction element 620, one can gain finer current control in the outputcurrent. This is because the quantized current in the lower loop (trace616 and element 620) is shared among a common wire of the upper loop(trace 602 and element 604) and lower loop. Thus the quantized currentin the lower loop I₂=n₂Φ₀/L₂ produces a small flux MI₂ in the upperloop, proportional to the shared wires mutual inductance M, that thenchanges the output current with finer control I=n₂Φ₀M/L₁L₂. In this way,the magnetic flux device enables even finer control over a second device650, such as a qubit or qubit coupler element.

As explained herein, if the predetermined criteria are not met by thefluxes applied to the SQUID 200, then the value of n does not change.For instance, if the applied fluxes fail to trace a closed-loop patharound a critical point corresponding to a current I=0, the value of nwill not change. Alternatively, if the applied fluxes fail to trace apath through a point where the phase of the magnetic flux control devicetransitions between positive and negative values as described herein,then the value of n will not change.

The process for changing the output state of the magnetic flux controldevice is near adiabatic. In some implementations, the magnetic fluxcontrol devices of the present disclosure can dissipate a factor of10³-10⁵ less power than CMOS-based or SFQ-based control devices. Thus,the magnetic flux control device of the present disclosure generatesvery little heat. Furthermore, in contrast to other qubit controldevices, such as SFQ-based control devices, the magnetic flux controldevices of the present disclosure do not require the use of on-chipdamping resistors. Because the power dissipation, and thus heatgeneration, of the magnetic flux control devices is so low, the magneticflux control devices can be arranged on or near the same chip as quantumcomputing circuit elements without substantially increasing the localchip temperature above a desired operating temperature (e.g., 20 mK),without causing transitions to undesired energy states, or withoutcontributing to decoherence.

The magnetic flux control devices of the present disclosure, such asdevices shown in FIGS. 1 and 6, can be used as control elements forapplying biases to one or more other devices, such as quantum circuitelements (e.g., quantum computing circuit elements). For example, insome implementations, a magnetic flux control device of the presentdisclosure can be used to apply a particular bias (e.g., a magnetic fluxsignal or a current signal) to a superconducting qubit (e.g., a fluxqubit, a phase qubit or a charge qubit). The bias from the magnetic fluxcontrol device can be used to initialize the superconducting qubit intoa particular desired state, to tune an operating frequency of asuperconducting qubit to a predetermined frequency, to tilt/perturb adouble well potential exhibited by a superconducting qubit, and/or toadjust a magnitude of a barrier between potential wells exhibited by asuperconducting qubit. For instance, by applying a first time-varyingmagnetic flux Φ₁ to a first sub-loop of the SQUID element of themagnetic flux control device, and a second time-varying magnetic flux Φ₂to a second sub-loop of the SQUID, in which the first time-varyingmagnetic flux Φ₁ and the second time-varying magnetic flux Φ₂ meetpredetermined criteria as explained herein, a bias supplied by themagnetic flux control device can be changed by a non-zero integermultiple n. For example, a magnetic flux signal or current signalgenerated by the magnetic flux control device can be incremented ordecremented by a non-zero integer multiple n of the flux quantum Φ₀,leading to a change in a state (e.g., operating frequency) of the qubit.

In some implementations, a magnetic flux control device of the presentdisclosure, such as the devices shown in FIGS. 1 and 6, can be used toapply a particular bias (e.g., a magnetic flux signal or a currentsignal) to a superconducting qubit coupling element. The bias from themagnetic flux control device can be used to increase or decrease thecoupling strength of the superconducting qubit coupling element, suchthat a level of coupling between two or more qubits can be modified. Forexample, by applying a first time-varying magnetic flux Φ₁ to a firstsub-loop of the SQUID element of the magnetic flux control device, and asecond time-varying magnetic flux Φ₂ to a second sub-loop of the SQUID,in which the first time-varying magnetic flux Φ₁ and the secondtime-varying magnetic flux Φ₂ meet predetermined criteria as explainedherein, a magnetic flux signal or current signal generated by themagnetic flux control device can be incremented or decremented by anon-zero integer multiple n of the flux quantum Φ₀, leading to acorresponding change in the coupling strength of the superconductingqubit coupling element. Alternatively, or in addition, the biasgenerated by the magnetic flux control device can be used to tune afrequency of the qubit coupler.

The magnetic flux control device according to the present disclosurerequires at least two input signals (e.g., one for each sub-loop of theSQUID) and only changes an output state (e.g., a current I or flux M)when the input signals meet predetermined criteria. In all othercircumstances, the output state of the device does not change.Accordingly, in some implementations, the magnetic flux control deviceis suitable for use as a logical AND gate. That is, only when inputsignals having the requisite predetermined criteria are applied to bothsub-loops of the SQUID can the effective phase offset of the devicechange by 2π, such that an effective flux through the device changes bya flux quantum Φ₀ (e.g., output a 1). In all other instances (i.e., whenonly one magnetic flux signal has the requisite time-varying magnitudeis applied or when no magnetic flux signals having the requisitetime-varying magnitude are applied), the effective flux does will notchange (e.g., output a 0).

In some implementations, multiple magnetic flux control devices can becombined with corresponding circuit elements (e.g., quantum circuitelements) in an array to provide controllable matrix addressing of thecircuit elements (e.g., the quantum circuit elements). FIG. 7 is aschematic illustrating an example of a controllable matrix array 700 ofmagnetic flux control devices. As shown in FIG. 7, the array 700includes multiple cells 702, 704, 706, and 708 arranged in rows andcolumns. Each cell 702, 704, 706, and 708 includes a magnetic fluxcontrol device, such as the magnetic flux control devices describedherein. For example, in some implementations, the magnetic flux controldevice includes a superconducting trace coupled to a SQUID having threenon-linear inductor junctions coupled in parallel. Each cell 702, 704,706, and 708 also includes a circuit element positioned adjacent to themagnetic flux control device. The circuit element can include, e.g., aqubit and/or a qubit coupling element. Other circuit elements are alsopossible.

The array 700 also includes multiple vertical control lines (e.g.,control line 712, 714). Each vertical control line extends along acorresponding column of the array 700 and is positioned next to eachcell within the column. For example, control line 712 is positioned nextto cells 702 and 706, whereas control line 714 is arranged next to cells704 and 708. The vertical control lines can be formed fromsuperconducting traces and are associated with correspondinginductances.

The array 700 also includes multiple horizontal control lines (e.g.,control line 716, 718). Each horizontal control line extends along acorresponding column of the array 700 and is positioned next to eachcell within the column. For example, control line 716 is positioned nextto cells 702 and 704, whereas control line 718 is arranged next to cells706 and 708. The horizontal control lines can be formed fromsuperconducting traces and are associated with correspondinginductances.

The vertical control lines (e.g., lines 712, 714) are coupled to (e.g.,electrically connected to) a column select generator 710. The horizontalcontrol lines (e.g., lines 716, 718) are coupled to (e.g., electricallyconnected to) a row select generator 720. Each of column selectgenerator 710 and row select generator includes circuitry (e.g., acurrent source) configured to generate a waveform that is applied to acontrol line. The column and row select generators can be configured toprovide a unique waveform to each control line to which the generator iscoupled. For example, in some implementations, the column selectgenerator can be programmed to deliver a unique waveform to eachvertical control line, whereas the row select generator can beprogrammed to deliver a unique waveform to each horizontal control line.

Thus, given an i×j matrix of cells, where each cell includes a circuitelement (e.g., a quantum circuit element such as a qubit or qubitcoupler element) and a corresponding magnetic flux control device, thenumber of connectors needed to address the circuit elements is i+j.Furthermore, the magnetic flux control device of each cell of the matrixserves to actively maintain the state of the circuit element while othercells are being addressed, thus preventing crosstalk from inadvertentlychanging the state of an unaddressed cell. That is, when input signalsmeeting the predetermined criteria as described herein are supplied to amagnetic flux control device, the magnetic flux control device is placedin a persistent output state (e.g., the magnetic flux control devicegenerates a persistent current) that does not change, until and unlessinput signals meeting the predetermined criteria again are applied tochange the output state. Accordingly, there is no need to providecontinuous application of input signals to a cell to maintain a desiredoutput state. Furthermore, because the control device can activelymaintain the state of the circuit element, the total power dissipationrequired to address the array is significantly lower than it would be ifeach cell had to be addressed continuously to maintain its state.

In some implementations, the magnetic flux control device of the presentdisclosure can be used as a multi-level memory device. As explainedherein, the current I through the magnetic flux device can be expressedas I=M/L, where M is the flux through the superconducting loop of thedevice and can be expressed as M=nΦ₀. With each 2π change in effectivephase offset of the magnetic flux control device, the integer value nchanges by 1, leading to a corresponding change in the output state(e.g., the current I or flux M) of the magnetic flux control device. Theoutput state of the magnetic flux control device does not change unlessthere is another 2π change in the effective phase offset of the device.Thus, the magnetic flux control device can be placed in variousdifferent output or “memory” states depending on the value of n. Thatis, the different values of I (or the different magnetic fluxes M)generated by the control device can represent different memory states,in which each memory state will be maintained until and unless theappropriate flux signals are applied to the magnetic flux control deviceto change n again.

FIG. 8 is a table that illustrates an example of different output statesthat can be established by a multi-level memory based on the magneticflux control device according to the present disclosure. As shown inFIG. 8, four different output states of a magnetic flux control device,such as device 100, are presented. Each output state is mapped tocorresponding bits of information. For example, the multi-level memorycan be programmed to be in one of four different memory states, e.g. n,n+1, n+2, and n+3 that correspond to binary values “00”, “01”, “10”, and“11”, respectively.

Writing to the multi-level memory includes applying flux signals to theSQUID element of the magnetic flux control device, in which the fluxsignals satisfy the predetermined criteria required for the change ineffective phase offset, Δφ, of the device to be an integer multiple of2π. The change in effective phase offset, Δφ, shown in the table of FIG.8 is evaluated from a first memory state, n. For instance, to change thememory state of the multi-level memory from a first memory state to asecond memory state (e.g., from “00” to “01”), flux signals are appliedto the device to change the effective phase offset by 2π. To change thedevice from the first memory state to a third memory state (e.g., from“00” to “10”), flux signals are applied to the device to change thephase by 4π. To change the device from the first memory state to afourth memory state (e.g., from “00” to “11”), flux signals are appliedto the device to change the effective phase offset by 6π. Similarly, tochange the device from the fourth memory state to the first memory state(e.g., from “11” to “00”), flux signals are applied to the device tochange the effective phase offset by −6π. To change the device from thefourth memory state to the second memory state (e.g., from “11” to“01”), flux signals are applied to the device to change the effectivephase offset by −4π. To change the device from the fourth memory stateto the third memory state (e.g., from “11” to “10”), flux signals areapplied to the device to change the effective phase offset by −2π.Although the device associated with FIG. 8 is represented as having onlyfour different memory states, other numbers of memory states also can beused. In some implementations, the limit to the number of memory statescan be defined based on what time period is acceptable for changing thephase of the memory device.

Implementations of the quantum subject matter and quantum operationsdescribed in this specification can be implemented in suitable quantumcircuitry or, more generally, quantum computational systems, includingthe structures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The term“quantum computational systems” may include, but is not limited to,quantum computers, quantum information processing systems, quantumcryptography systems, topological quantum computers, or quantumsimulators.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, e.g., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In some implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible. It isunderstood that quantum memories are devices that can store quantum datafor a long time with high fidelity and efficiency, e.g., light-matterinterfaces where light is used for transmission and matter for storingand preserving the quantum features of quantum data such assuperposition or quantum coherence.

Quantum circuit elements (also referred to as quantum computing circuitelements) include circuit elements for performing quantum processingoperations. That is, the quantum circuit elements are configured to makeuse of quantum-mechanical phenomena, such as superposition andentanglement, to perform operations on data in a non-deterministicmanner. Certain quantum circuit elements, such as qubits, can beconfigured to represent and operate on information in more than onestate simultaneously. Examples of superconducting quantum circuitelements include circuit elements such as quantum LC oscillators, qubits(e.g., flux qubits, phase qubits, or charge qubits), and superconductingquantum interference devices (SQUIDs) (e.g., RF-SQUID or DC-SQUID),among others.

In contrast, classical circuit elements generally process data in adeterministic manner. Classical circuit elements can be configured tocollectively carry out instructions of a computer program by performingbasic arithmetical, logical, and/or input/output operations on data, inwhich the data is represented in analog or digital form. In someimplementations, classical circuit elements can be used to transmit datato and/or receive data from the quantum circuit elements throughelectrical or electromagnetic connections. Examples of classical circuitelements include circuit elements based on CMOS circuitry, rapid singleflux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices andERSFQ devices, which are an energy-efficient version of RSFQ that doesnot use bias resistors.

Fabrication of the quantum circuit elements and classical circuitelements described herein can entail the deposition of one or morematerials, such as superconductors, dielectrics and/or metals. Dependingon the selected material, these materials can be deposited usingdeposition processes such as chemical vapor deposition, physical vapordeposition (e.g., evaporation or sputtering), or epitaxial techniques,among other deposition processes. Processes for fabricating circuitelements described herein can entail the removal of one or morematerials from a device during fabrication. Depending on the material tobe removed, the removal process can include, e.g., wet etchingtechniques, dry etching techniques, or lift-off processes. The materialsforming the circuit elements described herein can be patterned usingknown lithographic techniques (e.g., photolithography or e-beamlithography).

During operation of a quantum computational system that usessuperconducting quantum circuit elements and/or superconductingclassical circuit elements, such as the circuit elements describedherein, the superconducting circuit elements are cooled down within acryostat to temperatures that allow a superconductor material to exhibitsuperconducting properties. A superconductor (alternativelysuperconducting) material can be understood as material that exhibitssuperconducting properties at or below a superconducting criticaltemperature. Examples of superconducting material include aluminum(superconductive critical temperature of 1.2 kelvin) and niobium(superconducting critical temperature of 9.3 kelvin).

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. For example, the actions recited in the claims can be performedin a different order and still achieve desirable results. In certaincircumstances, multitasking and parallel processing may be advantageous.Moreover, the separation of various components in the implementationsdescribed above should not be understood as requiring such separation inall implementations.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made without departingfrom the spirit and scope of the invention. Accordingly, otherimplementations are within the scope of the following claims.

What is claimed is:
 1. A method comprising generating a bias signal froma first device, and applying the bias signal to a second device, thefirst device comprising (a) a superconducting trace and (b) asuperconducting quantum interference device (SQUID) having at leastthree non-linear inductor junctions coupled in parallel, wherein a firstterminal of the SQUID is electrically coupled to a first end of thesuperconducting trace, and a second terminal of the SQUID iselectrically coupled to a second end of the superconducting trace toform a loop, wherein generating the bias signal from the first devicecomprises: applying a first time-varying magnetic flux Φ₁ to a firstsub-loop of the SQUID; and applying a second time-varying magnetic fluxΦ₂ to a second sub-loop of the SQUID, wherein the first time-varyingmagnetic flux Φ₁ and the second time-varying magnetic flux Φ₂ areapplied such that a value of a superconducting phase of the first deviceis incremented or decremented by a non-zero integer multiple n of 2π. 2.The method of claim 1, wherein the output state comprises an effectivephase offset of a current through the first device.
 3. The method ofclaim 1, wherein the output state comprises an effective flux throughthe first device.
 4. The method of claim 1, wherein each of a maximummagnitude of the first time-varying magnetic flux Φ₁ and a maximummagnitude of the second time-varying magnetic flux Φ₂ is less than theflux quantum Φ₀.
 5. The method of claim 1, wherein a ratio, Φ₁/Φ₀, ofthe first time-varying magnetic flux Φ₁ to the flux quantum Φ₀, and aratio, Φ₂/Φ₀, of the second time-varying magnetic flux Φ₂ to the fluxquantum Φ₀ trace a path around a point where a current value through thefirst device is
 0. 6. The method of claim 5, wherein the ratio Φ₁/Φ₀ andthe ratio Φ₂/Φ₀ are approximately equal to 1/3 at the point where thecurrent through the first device is
 0. 7. The method of claim 5, whereinthe integer multiple n is incremented when the ratio Φ₁/Φ₀ and the ratioΦ₂/Φ₀ trace the path along a first direction around the point where thecurrent through the first device is 0, or wherein the integer multiple nis decremented when the ratio Φ₁/Φ₀ and the ratio Φ₂/Φ₀ trace the pathalong a second direction that is opposite to the first direction.
 8. Themethod of claim 5, wherein the path is a closed-loop path.
 9. The methodof claim 1, wherein a ratio, Φ₁/Φ₀, of the first time-varying magneticflux Φ₁ to the flux quantum Φ₀, and a ratio, Φ₂/Φ₀, of the secondtime-varying magnetic flux Φ₂ to the flux quantum Φ₀ trace a paththrough a point where an effective phase offset of the current throughthe first device is
 0. 10. The method of claim 9, wherein the path is aclosed-loop path.
 11. The method of claim 1, wherein applying the firsttime-varying magnetic flux and the second time-varying magnetic fluxcomprises changing the phase associated with each Josephson junction ofthe SQUID by 2π.
 12. The method of claim 1, further comprising coolingthe first device to below the superconducting critical temperature of asuperconducting material in the superconducting trace.
 13. The method ofclaim 1, wherein the first time-varying magnetic flux Φ₁ and the secondtime-varying magnetic flux Φ₂ are sequentially applied.
 14. The methodof claim 12, wherein the first time-varying magnetic flux Φ₁ and thesecond time-varying magnetic flux Φ₂ overlap in time.